One of the ideas behind planar near-field acoustic holography (PNAH) is to determine an inverse solution of the wave equation by only making use of planar acoustic information at a certain distance from the source of interest. In fact, hologram originated from the Greek word “holos”, meaning “whole”, and “gramma”, meaning “message”. In short, the hologram may contain all information of the acoustic source. However, this information may be hidden in the hologram, and data processing may be necessary to reconstruct information about the acoustic source.
In the mid 1980s the PNAH theory was first introduced. A regularization method was described in “Regularization in PNAH by means of L-curve”, by R. Scholte et al., in Proceedings of Forum Acusticum 2005, Budapest, Hungary. Some of the principles from the mentioned theory will be presented below.
A boundary condition which may be imposed in the application of PNAH is the absence of acoustic sources between the measurement plane (hologram) at z=zh and the reconstruction (source) plane at z=zs. The hologram may be built up from complex sound pressure measurements that take place in x,y-space. The complex sound pressure p(x,y,zh,ω) is then transformed to the wavenumber domain by 2-dimensional spatial Fourier transform,
      P    ⁡          (                        k          x                ,                  k          y                ,                  z          h                ,        ω            )        =            ∫              -        ∞            ∞        ⁢                  ∫                  -          ∞                ∞            ⁢                        p          ⁡                      (                          x              ,              y              ,                              z                h                            ,              ω                        )                          ⁢                  ⅇ                      -                          j              ⁡                              (                                                                            k                      x                                        ⁢                    x                                    +                                                            k                      y                                        ⁢                    y                                                  )                                                    ⁢                  ⅆ          x                ⁢                              ⅆ            y                    .                    
This operation results in a 2-dimensional wavenumber spectrum where two important regions can be discriminated: the propagating and evanescent wave region. The propagating wave region is positioned around wavenumber 0, bounded by a circle with radius k (radiation circle), where k depends on the sound-frequency of interest,
      k    =                  ω                  c          0                    ⁢                          [              rad        m            ]        ,with c0 the velocity of sound and ω the angular sound frequency. The inside region contains propagating waves, while the outside region contains evanescent or decaying waves. The decay as a function of distance for the evanescent waves is regarded exponential. The wavenumber spectrum at the hologram plane can be multiplied with the inverse of the exponential, also referred to as the propagator, resulting in the wavenumber domain at any desired distance z between source and hologram plane,P(kx,ky,z,ω)=P(kx,ky,zh,ω)e−jkz(z−zh).
This provides the wavenumber spectrum at the distance z. From this, the spatial pressure can be determined by applying the inverse 2-dimensional Fourier transform,
      p    ⁡          (              x        ,        y        ,        z        ,        ω            )        =            1              4        ⁢                  π          2                      ⁢                  ∫                  -          ∞                ∞            ⁢                        ∫                      -            ∞                    ∞                ⁢                              P            ⁡                          (                                                k                  x                                ,                                  k                  y                                ,                z                ,                ω                            )                                ⁢                      ⅇ                          j              ⁡                              (                                                                            k                      x                                        ⁢                    x                                    +                                                            k                      y                                        ⁢                    y                                                  )                                              ⁢                      ⅆ                          k              x                                ⁢                                    ⅆ                              k                y                                      .                              
Other propagators can be determined to transform hologram sound pressure or particle velocity to sound pressure, particle velocity and sound intensity at the source.
In practice it is difficult to continually measure the sound pressure on an infinitely wide plane. Instead, spatial sampling over a finite aperture is performed. This spatially sampled data is then fast Fourier transformed using the 2-dimensional FFT algorithm. However, the results of such algorithms are not always satisfying.